J. Flusser, T. Suk, and B. Zitová Moments and Moment Invariants in Pattern Recognition The slides accompanying the book

Copyright notice The slides can be used freely for non-profit education provided that the source is appropriately cited. Please report any usage on a regular basis (namely in university courses) to the authors. For commercial usage ask the authors for permission. The slides containing animations are not appropriate to print. © Jan Flusser, Tomas Suk, and Barbara Zitová, 2009

Contents 1. Introduction to moments 2. Invariants to translation, rotation and scaling 3. Affine moment invariants 4. Implicit invariants to elastic transformations 5. Invariants to convolution 6. Orthogonal moments 7. Algorithms for moment computation 8. Applications 9. Conclusion

Chapter 5

Invariants to convolution

Motivation – character recognition

Motivation – template matching

Image degradation models Space-variant Space-invariant  convolution

Two approaches Traditional approach: Image restoration (blind deconvolution) and traditional invariants Proposed approach: Invariants to convolution

Invariants to convolution for any admissible h

The moments under convolution Geometric/central Complex

PSF is centrosymmetric with respect to its centroid and has a unit integral Assumptions on the PSF supp(h)

Common PSF’s are centrosymmetric

Invariants to centrosymmetric convolution where (p + q) is odd What is the intuitive meaning of the invariants? “Measure of anti-symmetry”

Invariants to centrosymmetric convolution

Convolution invariants in FT domain

Relationship between FT and moment invariants where M(k,j) is the same as C(k,j) but with geometric moments

Template matching sharp image with the templates the templates (close-up)

Template matching a frame where the templates were located successfully the templates (close-up)

Template matching performance

Valid region Boundary effect in template matching

Invalid region - invariance is violated Boundary effect in template matching

zero-padding

Extension to n dimensions where |p| is odd

N-fold rotation symmetry, N > 2 Circular symmetry Gaussian PSF Motion blur PSF Other types of the blurring PSF The more we know about the PSF, the more invariants and the higher discriminability we get

Invariants to N-fold PSF If (p-q)/N is integer  the invariant is zero

Invariants to circularly symmetric PSF If p = q  the invariant is zero (except p = q = 0)

Invariants to Gaussian PSF If p = q = 1  the invariant is zero

Discrimination power The null-space of the blur invariants = a set of all images having the same symmetry as the PSF. One cannot distinguish among symmetric objects.

Recommendation for practice It is important to learn as much as possible about the PSF and to use proper invariants for object recognition

Combined invariants Convolution and rotation For any N

Satellite image registration by combined invariants

( v1 1, v2 1, v3 1, … ) ( v1 2, v2 2, v3 2, … ) min distance(( v1 k, v2 k, v3 k, … ), ( v1 m, v2 m, v3 m, … )) k,m Control point detection

Control point matching

Transformation and resampling

Combined blur-affine invariants Let I(μ00,…, μPQ) be an affine moment invariant. Then I(C(0,0),…,C(P,Q)), where C(p,q) are blur invariants, is a combined blur-affine invariant (CBAI).

Digit recognition by CBAI CBAI AMI